Bawylie
A very OK person
Counterpoint, if the player tells you the skill/tool/proficiency being used, **that also** brings with it the linkage to actions taken.
If the player says rolling perception that says looking, if they say check with tools, thats touching fiddling etc. Quite possibly DCs are different, definitely risks and outcomes different.
Thats where i get amused at the rigid "GM chooses checks, players shall not" when the go with how approach reveals type of check - it works both ways.
Both can work well, both can botch usually on the same ground - do the players and gm have the same understanding of what skills and tools and such cover or do they have different expectatiins.
How about if you don’t even use skills? Ability checks only doesn’t quite cover every approach without further clarification.
Use of thieves’ tools might, as you suggest, indicate a particular approach. Does a dex check? Int check?